Flow-independent parameter estimation based on tidal breathing exhalation profiles

ABSTRACT

Parametric characterization of nitric oxide (NO) gas exchange using a two-compartment model of the lungs is a non-invasive technique to characterize inflammatory lung diseases. The technique applies the two-compartment model to parametric characterization of NO gas exchange from a tidal breathing pattern. The model is used to estimate up to six flow-independent parameters, and to study alternate breathing patterns.

The application is related to U.S. provisional patent application Ser.No. 60/354,781 filed Feb. 5, 2002 and entitled FLOW-INDEPENDENTPARAMETER ESTIMATION BASED ON TIDAL BREATHING EXHALATION PROFILES, andto U.S. provisional patent application Ser. No. 60/380,175 filed May 13,2002 and entitled CHARACTERIZING NITRIC OXIDE EXCHANGE DYNAMICS DURINGTIDAL BREATHING, both of which are incorporated herein by reference andto which priority is claimed pursuant to 5 USC 119.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to a method and apparatus for determiningphysiological parameters indicative of lung condition, which parametersare independent of air flow rates based on nitrogen monoxide content inexhalation, which content is dependent on air flow rates.

2. Description of the Prior Art

It is known that nitric oxide (NO) is produced in the lung and appearsin the exhaled breath. The exhaled concentration is elevated inimportant inflammatory diseases such as asthma. A significant fractionof exhaled NO which is unique among endogenous gases comes from theairways. A single breath technique has been invented by one of theinventors and is the subject of a copending application in a priorapplication, which technique required inhalation to total lung capacityfollowed by a breath hold of up to 20 seconds followed by an exhalationwith a decreasing flow rate which lasted approximately 15 seconds. Thusapproximately 35 seconds was required to complete the maneuver. Thisduration is not possible for subjects who cannot follow directions(i.e., small children or subjects who are unconscious) or who havecompromised lung function.

BRIEF SUMMARY OF THE INVENTION

The present invention is a technique to characterize nitric oxideexchange dynamics in the lungs during tidal breathing and thusrepresents a potentially robust technique that a wide range of subjects,including children, can perform. The technique is a breathing maneuvercombined with a mathematical model, data filtering, and parameterestimation techniques to characterize the exchange dynamics of nitricoxide (NO) in the lungs.

In order to perform the technique, the subject breathes normally(tidally) into a mouthpiece while exhaled nitric oxide concentration andflow rate are recorded simultaneously. During the tidal breathing, thesubject inspires NO-free air and the soft palate is either closedvoluntarily or by applying a small negative pressure in the nasalcavity.

The exhaled NO concentration is separately simulated with atwo-compartment model. The first compartment corresponds to the airwaysand the second compartment corresponds to the aveolar regions of thelungs. After data filtering and nonlinear least squares regression onthe simulated exhaled NO concentration, the optimal values of three tosix parameters were obtained. An objective of the simulation was tocharacterize NO exchange dynamics in both the airways and the alveolarregions.

Changes in lung volume for tidal breathing (400–800 ml) are smaller thanfor single breath maneuvers. A single cycle (inhalation and exhalation)occurs over a relatively short time frame (4–8 sec.), and exhalationprofiles are observed in a narrow window (2–4 sec.). Over a singleexhalation, there is little time to accumulate appreciable amounts of NOin the airway and alveolar components. Hence, tidal breathing profilesare flatter and lack the easily recognizable characteristics of thesingle breath maneuvers of the accepted techniques of the recent past.Furthermore, expired NO levels for tidal breathing are roughly 4-foldlower (5–10 ppb) than those observed for single breath maneuvers. Yet,the present technique provides a way of characterizing the NO exchangedynamics while the subject breathes tidally. That is, the subjectbreathes slowly and normally while at rest. Thus, advantageously,essentially any subject who can breathe into a mouthpiece can performthe test maneuver with no special training or skills (i.e., breathholding).

The invention can be used to quantify NO exchange dynamics and thuscharacterize metabolic and structural features not characterized bycurrently accepted techniques. The technique of the present invention ispotentially most useful for longitudinally following subjects withinflammatory diseases. For example, a subject with asthma may breathtidally for two minutes while the NO concentration and flow rate aremonitored. Our mathematical model and parameter estimation techniqueswould then estimate the flow-independent NO parameters. The subject mayrepeat this maneuver at regular intervals thereafter. Changes in theparameters in subsequent test maneuvers might prompt therapeuticintervention. This technique is also useful in achieving additionalobjectives with subjects having inflammatory diseases, such asmonitoring the efficacy of the therapy during treatment.

In one aspect, the invention characterizes the same exchange dynamicswhile the subject breathes tidally, and in particular identifies theoptimal breathing pattern. Thus, although other breathing patterns maybe possible, they may require effort or time on the part of the subject.

As discussed above, nitric oxide (NO) is produced in the lung andappears in the exhaled breath. The exhaled concentration is elevated inimportant inflammatory diseases such as asthma. A significant fractionof exhaled NO arises from the airways, which NO is unique amongstendogenous gases. This aspect of the invention comprises a theoreticalanalysis to predict the optimum tidal breathing pattern to characterizenitric oxide exchange dynamics in the lungs.

Parametric characterization of nitric oxide (NO) gas exchange using atwo-component model of the lungs as discussed above is a potentiallypromising, non-invasive technique to characterize inflammatory lungdiseases. Until the advances of the present invention, this techniquewas limited to single breath maneuvers, including pre-expiratory breathhold, which is cumbersome for children and individuals with compromisedlung function. The present invention applies the two-compartment modelto parametric characterization of NO gas exchange from a tidal breathingpattern. This model's potential to estimate up to six flow-independentparameters. The model also aid in studying alternate breathing patterns,such as varying breathing frequency and inspiratory/expiratory flow rateratio at constant alveolar ventilation rate. We identify four easilycharacterized flow-independent parameters, which include maximum airwayflux, steady state alveolar concentration, airway volume, and deadspacevolume (uncertainty<10%). Rapid inhalation followed by slow (longduration) exhalation as well as increasing the number of observed tidalbreaths facilitates estimates of all flow independent parameters. Theresults demonstrate that a parametric analysis of tidal breathing datacan be utilized to characterize NO pulmonary exchange.

One embodiment of the method of characterizing nitric oxide exchange inlungs during tidal breathing further comprises the steps of:representing estimated values by a data curve or an equation definingthe curve; wherein said step of estimating comprises determining a firstconfidence interval after a first monitoring time; and wherein said stepof estimating further comprises projecting at least one of said valueshaving improved accuracy and an improved second confidence intervalafter a second predetermined monitoring time, wherein the projection isbased on the data curve or the equation representing the curve.

In another embodiment the step of estimating further comprises improvingthe accuracy of the parameter values by reducing the frequency of tidalbreathing during monitoring. The frequency of breathing is reduced byincreasing the time for each tidal breath to in the range from 7 to 17seconds. The step of estimating further comprises improving the accuracyof the parameter values by increasing the flow rate ratio q of theinhalation flow rate Q_(I) relative to the exhalation flow rate Q_(E).

In still another embodiment the method further comprises increasing theflow rate ratio q into the range from 6/5 to 12.

While the apparatus and method has or will be described for the sake ofgrammatical fluidity with functional explanations, it is to be expresslyunderstood that the claims, unless expressly formulated under 35 USC112, are not to be construed as necessarily limited in any way by theconstruction of “means” or “steps” limitations, but are to be accordedthe full scope of the meaning and equivalents of the definition providedby the claims under the judicial doctrine of equivalents, and in thecase where the claims are expressly formulated under 35 USC 112 are tobe accorded full statutory equivalents under 35 USC 112. The inventioncan be better visualized by turning now to the following drawingswherein like elements are referenced by like numerals.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a graph showing a comparison between the model and theexperimental data.

FIG. 1B is a graph similar to FIG. 1A having a different breath profile.

FIG. 2A is a detailed graph of the third breath of FIG. 1A and includingfiltered results.

FIG. 2B is a detailed graph similar to FIG. 2A showing a special case offiltering.

FIG. 3 is a table summarizing the data shown in FIGS. 1A and 2A.

FIG. 4 is a graph showing the reduction in parameter uncertainty withthe number of tidal breaths monitored.

FIG. 5 is a schematic representation of the two-compartment modelwherein exhaled nitric oxide has sources in both the alveolar(J_(alvNO)) and the airway (J_(awNO)) regions of the lungs, whichrepresent the two compartments.

FIG. 6A is a graph depicting how the flow rate profile can be generallyrepresented by a square wave in which the flow rates during inspirationand expiration are assumed constant.

FIG. 6B is a schematic representation of a single tidal breath flow rateprofile.

FIG. 6C is a schematic representation of a single tidal breathexhalation profile.

FIG. 7 is a table showing central, upper, and lower limits in accordancewith the graph of FIG. 7 of the flow-independent parameter valuesJ′_(awNO), D_(awNO), {circumflex over (D)}_(alvNO), C_(alv,ss), V_(aw),and V_(ds).

FIG. 8 is a graph showing fractional uncertainties, Δx_(j), of theflow-independent parameters, J′_(awNO), D_(awNO), C_(alv,ss),{circumflex over (D)}_(alvNO) V_(aw), and V_(ds), for one minute oftidal breathing with q=Q_(I)/Q_(E)=t_(E)/t_(I)=2 and f_(B)=12 min⁻¹.

FIGS. 9A–9F are graphs showing the influence of each of theflow-independent parameters, respectively, upon the exhalation profile,C_(E)(t), for a single tidal breath with q=:Q_(I)/Q_(E)=t_(E)/t_(I)=2and f_(B)=12 min⁻¹ in accordance with FIGS. 7 and 8;

FIG. 10 is a graph showing dependence of the fractional uncertainties,Δx_(j), upon total monitoring time with all flow-independent parametersat their central values, and q=Q_(I)/Q_(E)=t_(E)/t_(I)=2 and f_(B)=12min⁻¹.

FIGS. 11A–11C are graphs showing dependence of the fractionaluncertainties, Δx_(j), upon the flow rate ratio,q=Q_(I)/Q_(E)=t_(E)/t_(I), and breathing frequency, f_(B) (min⁻¹) forthe flow-independent parameters D_(awNO), C_(alv,ss), and {circumflexover (D)}_(alvNO) at their central values.

The invention and its various embodiments can now be better understoodby turning to the following detailed description of the preferredembodiments which are presented as illustrated examples of the inventiondefined in the claims. It is expressly understood that the invention asdefined by the claims may be broader than the illustrated embodimentsdescribed below.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Single breath maneuvers can distinguish inflammatory diseases, such asasthma, cystic fibrosis, and allergic alveolitis, using either exhaledconcentration alone, or more positional specific parametriccharacterization, such as airway diffusing capacity (D_(awNO)), maximumairway flux (J′_(awNO)) and steady state alveolar concentration(C_(alv,ss)). Single breath maneuvers may be difficult to perform,especially for children and individuals with compromised lung function.Hence the utility of tidal breathing to characterize NO exchange hasbeen explored and is described herein.

Analysis of tidal breathing exhalation profiles presents new challengesrelative to single breath maneuvers, such as smaller changes in lungvolume and a shorter duration for exhalation. The shorter duration ofexhalation reduces accumulation of NO in the airway compartment,resulting in expired NO levels, which are more than four-fold lower thanthose observed for single-breath maneuvers. Analyzing multipleconsecutive tidal breaths potentially offsets this limitation.

The two-compartment model for tidal breathing includes both inhalationand exhalation intervals, as well as the time dependence of the alveolarconcentration during exhalation. The initial alveolar concentration (atthe start of the first full exhalation) is estimated by assuming allprevious (unobserved) cycles (breaths) to be identical and consistentwith the first cycle. Axial diffusion and detailed airway geometry arenot included in the current version of the model. In addition,inhalation and exhalation flow rate profiles are approximated asconstant over their respective intervals. The current version of thetwo-compartment model predicts a discontinuity in the expired NO profileat the end of airway evacuation, where the alveolar phase of exhalationbegins. This discontinuity is most likely an artifact resulting from theassumptions of negligible axial diffusion, simplified airway geometryand ideal flow in the simulated system. If necessary, axial diffusion,more complex airway geometry, and less ideal flow can be incorporatedinto the model after its preliminary assessment using this simplermodel.

Two typical NO concentration profiles, each from 12 tidal breaths of anactual human subject, are illustrated in FIGS. 1A and 1B as identifiedby the peaks in NO against time. The subject was experienced atperforming breathing maneuvers. Hence, minimal nasal contamination canbe assumed. The observed data was sampled at 50 Hz and fitted to themodel. FIGS. 1A and 1B compare the experimental data (indicated by x's10) with the exhalation profiles predicted by the fitted two-compartmentmodel (indicated by a solid line 15). The data set illustrated in FIG.1A corresponds to normal tidal breathing (2,000–3,000 ml lung volume).In FIG. 1B, the first three breaths correspond to normal tidalbreathing. However, breaths 4–10 of FIG. 1B correspond to higher lungvolume (3,500–4,000 ml), with a prolonged expiration in the eleventhbreath. Elevated levels of exhaled NO in breaths 4–11 are evident oninspection of FIG. 1B.

FIG. 2A compares the experimental data (represented by x's 10) for thethird breath of FIG. 1A with the exhalation profile predicted by thefitted two-compartment model (solid line 15). Expired NO concentrationsare close to the lower detection limit. At these low levels, baselinefluctuations and noise, originating within analytical instrumentation,cause significant data scatter. Hence, uncertainties in observed NOconcentrations significantly limit how close the parameter estimatescome to the experimental data. Consequently, several different sets ofparameter values are generally consistent with these data, although ahigh degree of uncertainty is associated with the parameter estimates.

Several techniques are available to minimize uncertainty in parameterestimates resulting from data scatter. The appropriate solution dependsupon the source(s) of the error. During data acquisition, electronicnoise (e.g., electromagnetic interference, smearing, aliasing, amplifierdistortion, etc.) may be added to the signal. If the noise consistsprimarily of high frequency components, the signal may be filtered toresolve the underlying exhalation profile. This approach is called “lowpass filtering”, and the frequency at which the signal power isattenuated by 50% is referred to as the resolution bandwidth.

The dashed lines 20, 25 in FIG. 2A depict filtered data at resolutionbandwidths of 1 and 2 Hz, respectively, and the filtered data showsubstantial improvement in their correlation with the fittedtwo-compartment model. This technique is useful for identification ofdata acquisition problems, and can increase the degree of confidence inparameter estimates, provided it is justified. However, since the truefrequency spectrum of the signal is not known in advance, low passfiltering may inadvertently remove subtle features of the exhalationprofile and is rigorously applicable only when the signal iscontaminated by known systematic errors. In fact, spectral analysis ofthe NO analyzer baseline suggests that the dependence of noise amplitudeupon frequency is subtle, with a broad spectrum typical for random(“white”) noise rather than systematic errors.

In this case, a more efficacious approach is the Kalman Filteralgorithm, which estimates the most probable exhalation profiles basedon comparison of experimental observations with the model. It alsoforecasts probable values of physically meaningful systemcharacteristics, called the state variables, which may not be directlyobservable. The state estimation feature of the Kalman Filter ispotentially useful as a diagnostic tool to assess conditions withinpulmonary tissue. Herein we consider only the basic Kalman Filter, whichis applied for “on-line” removal of random error.

Kalman filtered data (indicated as a dot-dash line 30 in FIG. 2B) isreasonably consistent with the fitted two-compartment model over theairway and alveolar evacuation phases of exhalation. However, during theairway evacuation phase, the exhalation profile predicted by the modellies above the filtered data, which implies systematic deviation fromthe model. Sinusoidal deviation from the model is evident during thealveolar evacuation phase. This may result from systematic errors (e.g.,electronic noise, inadequacies in the model, etc), random errors(“white-noise”), or a combination of both.

We estimated six parameters (D_(awNO), J′_(awNO), {circumflex over(D)}_(alvNO),C_(alv,ss), V_(aw) and V_(ds)) by minimizing a “scorefunction”, comprised of the sum of two appropriately scaled,mean-squared error terms. Both of these terms are fractions of thedifferences between observed and predicted NO concentrations for each ofthe 12 tidal breaths. The first term is the sum of thesquared-differences between each sampled exhalation concentration andthe model prediction. The second term compares the total amount of NOexpired for each breath, as computed from the experimental data, withthat predicted by the model. Selection of a score function should bemade to reflect the most appropriate criterion for robust parameterestimation.

Predicted exhalation profiles are highly sensitive to the airway volume,V_(aw), and the dead space volume, V_(ds). Several parameter sets whichfit the data, were found with V_(aw), and V_(ds) in the ranges: 180–220and 20–50 ml, respectively. To simplify the statistical analysispresented herein, we fixed V_(aw) and V_(ds) at 200 ml and 40 ml,respectively. Hence, uncertainties were determined only for fourparameters (D_(awNO), J′_(awNO), {circumflex over (D)}_(alvNO),C_(alv,ss)).

Parameter estimates from the data depicted in FIGS. 1A and 2A aresummarized in the table of FIG. 3, based on a minimum squared-error fitof the model with the data observed from 12 tidal breaths. We performeduncertainty analysis to compute 90% confidence intervals based on ajoint parameter hypotheses. The unfiltered parameter estimates 35depicted in the table of FIG. 3 show reasonable precision for J′_(awNO)and {circumflex over (D)}_(alvNO). The parameter estimates for bothshown in Range column 40 were within ±20% at 90% confidence limits after12 breaths. However, large uncertainties are associated with D_(awNO)and C_(alv,ss). By extrapolating the available data, we estimated thenumber of tidal breaths required to achieve parameter uncertaintieswithin 5, 10, 25 and 50% of their mean values in column 45 (see the lastfour columns 50 of the table of FIG. 3). These estimates are based onjoint parameter hypotheses at 90% confidence. Thus, J′_(awNO) is easilycharacterized since its uncertainty is reduced to within 5% in only twobreaths. However, to achieve the same precision, 120 breaths (more than1,500 sampled exhalation concentrations) are required to characterize{circumflex over (D)}_(alvNO), and 1,100 breaths are required forC_(alv,ss). However, 12 breaths correspond to only about 2 minutes ofsampling time. Thus, after 7 minutes (roughly 40 breaths) theuncertainty of C_(alv,ss) is reduced to within 25%, and to within 10%after 45 minutes (260 breaths). Extrapolation of the unfiltered datasuggests that D_(awNO) can not be estimated for in the unfilteredscenario, irregardless of how many tidal breaths are monitored.

Reducing the parameter uncertainty was attempted by correlating themodel with Kalman filtered data, which is shown by the parameterestimates 55. In principle the Kalman filter reduces “white noise”,which in turn can reduce the “leverage effect” of outliers on parameterestimates. This method resulted in an increase in the correlationcoefficient (r²-value) from 0.50 to 0.87 (see dot-dash line in FIG. 2B)and significant reduction in the 90% confidence intervals for parameterestimates (see Kalman filtered results in the table of FIG. 3). Thismethod also reduces the required number of tidal breaths to achieve thevarious parameter uncertainties ranges (see the last four columns 50 ofthe table of FIG. 3). For example, in contrast to the unfiltered results35, only 8 breaths (less than 2 minutes) are required to determine{circumflex over (D)}_(alvNO)/V_(alv) at a precision of ±10% in thefiltered results 55. Although to achieve the same precision for D_(awNO)and C_(alv,ss) would require 5,400 breaths (about 15 hours) and 250breaths (about 45 minutes), respectively, these results are encouragingsince both random and systematic errors can be reduced further byapplying more advanced versions of the Kalman Filter.

The reduction in parameter uncertainty with the number of tidal breathsmonitored is illustrated graphically in FIG. 4 which is a graph whichshows for D_(NO,air), J_(NO,max), D_(NO,air)/V_(alv) and C_(alv,ss), thepercent deviation from mean at 90% confidence for one breath, 12 breathsand 100 breaths, which is based on the joint parameter hypotheses at 90%confidence for both the unfiltered (UF) and the Kalman filtered (KF)correlations. Percentiles for 100 tidal breaths were extrapolated fromthe data for 12 tidal breaths and are shown at 60 in FIG. 4. After 100breaths (roughly 20 minutes), the uncertainties of J′_(awNO),{circumflex over (D)}_(alvNO)/V_(alv) and C_(alv,ss) are reduced toreasonable levels. However, D_(awNO) remains poorly characterized. It isevident that D_(awNO) and C_(alv,ss) exhibit skewed distributions.Although this may be an artifact of the data extrapolation, it alsosuggests systematic deviation of the model in terms of its dependence onthese parameters.

In the case of systematic deviation of the model from the observedexhalation profiles, more accurate parameter estimates can be achievedby upgrading the model to incorporate some of the additional featuresdescribed above. On the other hand, if specific sources of electronicnoise are identified, techniques, such as the low pass filtering schemedescribed above, can be applied. The Kalman Filter is the best way tominimize the adverse effect of random error. Finally, if all of theabove sources of error are important, more advanced versions of theKalman Filter are available to provide smoother estimates by combiningthe basic algorithm with other concepts, such as spectral analysis andthe Principle of Maximum Likelihood. By using this “hybrid” approach,the most likely sources of error can be identified, and the mostprobable parameter estimates and their uncertainties can thereby beobtained.

Implementation of the Analysis

We have two primary goals. First, we explore the feasibility ofestimating six flow-independent parameters, characteristic of NO gasexchange during tidal breathing, by fitting the two-compartment model torepresentative experimental tidal breathing data. In this assessment, weassume some knowledge of the extent of random noise introduced intoexperimental tidal breathing data, which result from the limitations ofa typical analytical monitoring system. Our ultimate goal is to obtainestimates of the following flow-independent parameters from experimentaltidal breathing data: airway diffusing capacity (D_(awNO)), maximumvolumetric airway flux (J′_(awNO)), steady state alveolar concentration(C_(alv,ss)), alveolar diffusing capacity per unit alveolar volume({circumflex over (D)}_(alvNO)), and the airway compartment and deadspace volumes, V_(aw) and V_(ds), respectively.

Second, we explore a range of physiologically relevant tidal breathingpatterns, and identify the pattern(s) that minimizes the uncertainty inparameter estimates per unit sampling time.

Model Structure and Assumptions

We utilize the previously described two-compartment model with minormodifications. Only the salient features of the model and itsmodifications are presented here. As shown in FIG. 5, we model theconducting airways (i. e., the trachea and the first 17 airwaygenerations) and the respiratory bronchioles/alveolar region(generations 18 and beyond) as rigid and flexible compartments 65 and70, respectively. These compartments together form the airspace 72. Thebronchial mucosa 75 and alveolar membrane 80, respectively, provideairway tissue and alveolar tissue and surround the airway and alveolarcompartments 65 and 70. Together, these tissues make up the exteriorpulmonary tissue 82. NO is consumed by hemoglobin and other substratespresent in pulmonary blood vessels, such that the NO concentration iszero at the exterior pulmonary tissue boundary 85 (located distal to theair space). Arrows 90 and 95 indicate net fluxes of endogenouslyproduced NO, which diffuse into the air space 72 of the airway andalveolar compartments 65 and 70, are denoted J_(awNo) and J_(alvNO),respectively. On exhalation, NO is transported to the mouth by theconvection of the bulk air stream, where it appears in expired breath.

Airway Region

A differential mass balance for the airway compartment 65 describes NOconcentration in the airway gas space, C_(air)=C_(air)(t,V), as afunction of time, t, and axial position in units of cumulative volume,V, as derived in previous work. The airway compartment 65 is modeled asa cylinder with total volume V_(aw), and axial diffusion is neglected.Thus, for both inhalation and exhalation:

$\begin{matrix}{\frac{\partial C_{air}}{\partial t} = {{{- Q}\frac{\partial C_{air}}{\partial t}} + {{\hat{D}}_{awNO}\left\lbrack {C_{awNO} - C_{air}} \right\rbrack}}} & \text{(Equation 1)}\end{matrix}$where Q is the volumetric flow rate of air (Q=Q_(I)(t) for inhalationand Q=Q_(E)(t) for exhalation). The net flux of NO into the airway isapproximated as a linear function of C_(air) as shown previously,J_(awNO)=J′_(awNO)−D_(awNO) C_(air)=D_(awNO) [C_(awNO)−C_(air)], whereD_(awNO) is the airway diffusing capacity, J′_(awNO) is the maximumvolumetric airway flux, C_(awNO)=J′_(awNO)/D_(awNO) is the equivalentgas phase airway tissue concentration, and {circumflex over(D)}_(awNO)=D_(awNO)/V_(aw).

We assume that the initial condition for each exhalation is the finalcondition of the preceding inhalation, and the converse. From FIG. 5,appropriate boundary conditions are: C_(air)(t,V=O)=C_(alv)(t), forexhalation, and C_(air)(t,V=V_(aw)+V_(ds))=0, for inhalation (NO freeair source). For arbitrary flow rate profiles, a rigorous solution toEquation 1 above is obtained in terms of airway residence timefunctions, as described previously. Herein, we approximate theinhalation and exhalation flow rate profiles, Q_(I)(t) and Q_(E)(t), assquare-wave functions, represented by their respectivetime-weighted-averages, Q_(I) and Q_(E), over each time interval asdepicted in the graphs of FIGS. 6A and 6B. Thus, C_(air)(t,V), can beexpressed as an array of algebraic expressions within the airway(O<V<V_(aw)) and the dead space regions (V_(aw)<V<V_(aw)+V_(ds)), overthe inhalation and exhalation time intervals (see the Model Solution forSquare-wave Flow Rate Profiles section below). The exhalation profile,C_(E)(t), is determined by evaluating C_(air)(t,V=V_(aw)+V_(ds)) on theexhalation time interval.

Alveolar Region

A differential mass balance for NO in a well-mixed alveolar compartment,valid for both inhalation and exhalation was derived. The timedependence volume of NO concentration in the alveolar gas space,C_(alv)(t), is governed by:dC _(alv) /dt={circumflex over (D)} _(alvNO) [C _(alv,ss) −C _(alv)]−Q(C _(air,end) −C _(alv))/V _(alv)(t)  (Equation 2)where C_(air,end)−C_(air)(t,V=0) for inhalation, andC_(air,end)=C_(alv)(t) for exhalation. The alveolar volume, V_(alv)(t),is related to the flow rate by: dV_(alv)/dt=−Q_(I) where Q=−Q_(I) forinhalation and Q=Q_(E) for exhalation (see FIG. 5). We approximate thenet flux of NO into the alveolar tissue gas space as:J_(alvNO)=J′_(alvNO)−D_(alvNo)C_(alv)(t)=D_(alvNO)[C_(alv,ss)−C_(alv)(t)],where J′_(alvNO) is the maximum volumetric flux of NO into the alveolarcompartment and C_(alv,ss) is the steady state, mixed alveolarconcentration. D_(alvNO) is the diffusing capacity of NO in the alveolarregion, which we express per unit alveolar volume as: {circumflex over(D)}_(alvNO)=D_(alvNO)/V_(alv).

Previous work has demonstrated that C_(alv)(t) approaches C_(alv,ss) forbreath-hold times exceeding 10 s. However, for tidal breathing, we mustdetermine the time dependence of C_(alv)(t). Other studies have shownthat D_(alvNO) is roughly proportional to (V_(alv))^(2/3). Thus,alveolar diffusing capacity per unit alveolar volume, {circumflex over(D)}_(alvNO), is proportional to (V_(alv))^(−1/3). A rough sensitivityassessment implies that the percent variation in {circumflex over(D)}_(alvNO) is roughly one-third of the relative change in tidalvolume, ΔV_(alv)/V_(alv). Thus, for a typical tidal breath, whereΔV_(alv)/V_(alv) is 15%, we expect only 5% variation in {circumflex over(D)}_(alvNO), which is comparable to current experimental estimates.Herein, we assume {circumflex over (D)}_(alvNO) is a constant(flow-independent) parameter. Systematic errors, resulting from thisassumption increase in significance as ΔV_(alv)/V_(alv) increases.

Model Solution for Identical Breaths

All of our analysis assumes that each breath is identical and a dynamicsteady state is maintained in vivo. This results in a periodicexhalation profile. However, this is usually not observed in practice. Amore general solution, which allows the flow rates and time intervals ofinhalation and exhalation to vary with each breath, can be derived tomodel actual tidal breathing data (see the Model Solution forSquare-wave Flow Rate Profiles section below). For identical breaths, wemodel only the first observed breath, and denote the flow rates and timeintervals as simply Q_(I), Q_(E), t_(I) and t_(E), for inhalation andexhalation, respectively (see FIGS. 6A and 6B). Square-wave flow rateprofiles are shown in FIG. 6A. FIG. 6B shows one period of thesquare-wave profile of FIG. 6A. The residence times for the airway (a)and dead space (ds) compartments are: τ_(Ea)=V_(aw)/Q_(E) andτ_(Eds)=V_(ds)/Q_(E) (exhalation), and τ_(Ia)=V_(aw)/Q_(I) andτ_(Ids)=V_(ds)/Q_(I) (inhalation). We assume that the tidal volumechange, ΔV_(alv), exceeds the sum of the airway and dead space volumes,V_(aw)+V_(ds) (i.e., t_(E)>τ_(Ea)+τ_(Eds) and t_(I)>τ_(Ia)+τ_(Ids)).Integration of Equation 1 yields an analytical solution, for the NOconcentration profiles, C_(air)(t,V), which allows us to express theexhalation profile, C_(E)(t)=C_(air)(t,V=V_(aw)+V_(ds)), in terms of theinitial alveolar concentration, C_(alv)(t=t₀+t₁)=C_(alv,I).

The exhalation profile is divided into the classic three phasesrepresenting the deadspace (Phase I), and airway compartment (Phase II),and the alveolar compartment (Phase III). The profile shown is onlyrepresentative of the NO exhalation profile during tidal breathing; theprecise shape of the exhalation profile for NO depends on the values ofthe flow-independent parameters as shown in FIGS. 9A–9F and discussedbelow.

For identical breaths, C_(E)(t) reduces to:

${\overset{\_}{S}}_{j}^{{sr},{r\; m\; s}}$

Phase I: t_(I)≦t<t_(I)+τ_(Eds):C _(E)(t)=0  (Equation 3)

Phase II: t_(I)+τ_(Eds)≦t<t_(I)+τ_(Eds)+τ_(Ea):C _(E)(t)=C _(awNO)[1−e ^((−{circumflex over (D)}) ^(awNO) ⁽1+q ⁻¹^()(t−t) ^(I) ^(−τ) ^(Eds) ⁾]  (Equation 4)

Phase III: t_(I)+τ_(Eds)+τ_(Ea)≦t<t_(I)+t_(E):C _(E)(t)=C _(awNO) +[C _(alv,ss) −C _(alvNO) ][e^((−{circumflex over (D)}) ^(awNO) ^(τ) ^(Eds) ⁾]+(C _(alv,I) −C_(alv,ss))e ^((−{circumflex over (D)}) ^(awNO) ^(τ) ^(Eds)^(−{circumflex over (D)}) ^(alvNO) ^((t−t) ^(I) ^(−τ) ^(Eds) ^(−τ) ^(Ea)⁾⁾  Equation 5where q=Q_(I)/Q_(E)=t_(E)/t_(I). The shape of a typical exhalationprofile is depicted graphically in FIG. 6C. Phase I (Equation 3)corresponds to the exhalation of NO free air from the dead space region.Phase II (Equation 4) corresponds to the exhalation of air, originatingfrom the airway compartment at the start of exhalation. Phase III(Equation 5) describes the exhalation of air, originating from thealveolar compartment, which passes through the airway and dead spacecompartments on its way to the mouth (see FIG. 6C).

For the alveolar region, we denote the final conditions of eachinhalation and exhalation with the subscripts, E and I, respectively.Thus, for the first observed inhalation (t=0 to t_(I)), Equation 2 issubject to the initial conditions: C_(alv)(t=0) C_(alv,E) andV_(alv)(t=0)=V_(alv,E). Similarly, for exhalation (t=t_(I) tot_(I)+t_(E)), Equation 2 is subject to the initial conditions:C_(alv)(t=t_(I))=C_(alv,I) and V_(alv)(t=t_(i))=V_(alv,I). Forsquare-wave flow rate profiles the alveolar volume is given byV_(alv)(t)=V_(alv,E)+Q_(I)t for inhalation andV_(alv)(t)=V_(alv,I)−Q_(E) (t−t_(I)) for exhalation. An algebraicexpression for C_(alv,I) is derived by direct integration of Equation 2over the previous inhalation and exhalation time intervals:C _(alv,I) =Q _(I) [f _(t) C _(awNO) −f _(a) C _(alv,ss) ]/[V _(alv,I)−V _(alv,E) e ^(({circumflex over (D)}) ^(alvNO) ^((t−t) ^(I) ^(−τ)^(Eds) ^(−τ) ^(Ea) ⁾⁾ −Q′ _(I) f _(QI) ]+C _(alv,ss)  (Equation 6)where f_(t), f_(a) and f_(QI) are functions of D_(awNO), {circumflexover (D)}_(alvNO), V_(aw), V_(ds), Q_(I), Q_(E), t_(I) and t_(E), asdefined in the Nomenclature and Abbreviations section below. The aboveassumptions imply that the initial alveolar concentration, C_(alv,I, 1),for the first observed exhalation is the same as that for the previous(unobserved) exhalation, C_(alv,I,0) (see the Model Solution forSquare-wave Flow Rate Profiles section below).Model Solution for Square-wave Flow Rate Profiles

An analytical solution can be derived for Equations 1 and 2 forsquare-wave flow rate profiles and a general breathing pattern. We indexeach breath by the subscript, m, which starts at m=1 for the firstobserved breath (m=0 for the previous, unobserved breath). Eachinhalation begins at time t_(0,m)=Summation from i=1 to m−1 of[t_(I,i)+t_(E,i)], where t_(0,1)=0 at the start of the first observedinhalation. Thus, Q_(I,m) and Q_(E,m), represent the inhalation andexhalation flow rates averaged over their respective time intervals,t_(I,m) and t_(E,m) (e. g., Q_(I,1) and Q_(E,1) on the time intervals,t=0 to t_(I,1), and t_(I,1) to (t_(I,1)+t_(E,1)), respectively, for thefirst observed breath).

For breath, m, we define the residence times of the airway (a) and deadspace (ds) compartments, for inhalation (I) and exhalation (E), asτ_(Ia,m)=V_(aw)/Q_(I,m), τ_(Ids,m)=V_(ds)/Q_(I,m),τ_(Ea),m=V_(aw)/Q_(E,m), and τ_(Eds,m)=V_(ds)/Q_(E,m), respectively.Integration of Equation 1 yields an analytical solution for the NOconcentration profiles, C(t,V), within the airway (V=0 to V_(aw)) andthe dead space regions (V=V_(aw) to V_(aw)+V_(ds)), from which we obtainthe exhalation profile, C_(E)(t)=C_(air)(t,V=V_(aw)+V_(ds)) on theexhalation time interval, t=t_(0,m)+t_(I,m) to t_(0,m+1,). This solutionis analogous to those for Equations 3 to 5, which are omitted forbrevity.

The initial, alveolar region conditions, for inhalation and exhalation,are equated to the final conditions of each exhalation and inhalation(denoted by the subscripts E and I), respectively. Thus, C_(alv)(t) andV_(alv)(t) are evaluated as: C_(alv,E,m−1), V_(alv,E,m−1), C_(alv,I,m)and V_(alv,I,m) at: t=t_(0,m) and t_(0,m)+t_(I,m), respectively. For thegeneral case, we assume the ratio, q₁=Q_(I,1)/Q_(E,1)=t_(E,1)/t_(I,1),for the first observed breath is identical to all previous (unobserved)breaths, which implies that the initial alveolar concentration,C_(alv,I,1), for the first observed exhalation is identical to that forthe previous (unobserved) exhalation, C_(alv,I,0). Finally, we relatethe initial conditions for consecutive exhalations, C_(alv,m) andC_(alv,I,m−1):C _(alv,I,m)=(C _(alv,I,m−1) −C _(alv,ss))[V _(alv,E,m−1) e^(^)(−{circumflex over (D)} _(alvNO)(t _(I,m) +t _(E,m−1)))+Q _(I,m) f_(QI,m) ]/V _(alv,I,m) +Q _(I,m) [f _(t,m) C _(awNO) −f _(a,m) C_(alv,ss) ]/V _(alv,I,m) +C _(alv,ss)  (Equation 7)where f_(t,m), f_(a,m) and f_(QI,m) are defined in analogous fashion tof_(t), f_(a) and f_(QI), respectively (see the Nomenclature andAbbreviations section at the end of the detailed description). For thespecial case of “dynamic steady state” (i.e., identical breaths) theexhalation profile is periodic, with period, t_(I)+t_(E), and we maydrop the subscript, m, for: Q_(I,m), Q_(E,m), t_(E,m), t_(I,m),V_(alv,E,m), V_(alv,I,m), C_(alv,E,m), C_(alv,I,m), f_(t,m), f_(a,m),and f_(QI,m). Thus, for identical breaths, Equation 7 reduces toEquation 6.Confidence Intervals

We computed theoretical confidence intervals (uncertainties) forhypothetical estimates of the six flow-independent parameters (definedabove) from experimental tidal breathing data using the two-compartmentmodel: D_(awNO), J′_(awNO), C_(alv,ss), {circumflex over (D)}_(alvNO),V_(aw), and V_(ds). In practice, {circumflex over (D)}_(alvNO), V_(aw)and V_(ds), are usually specified, based on previous experiments,morphology, and sampling system characteristics. However, we assess theefficacy of estimating these additional parameters from exhalationprofile data, herein.

The Sensitivity an Uncertainty Analysis section below describes themethodology used to compute theoretical 90% confidence intervals for theestimated flow-independent parameters, X_(j). We express our results interms of the fractional uncertainties, Δx_(j) (indexed by j=1, 2, . . ., P=6), or the fractional deviation from the “unbiased” or centralvalue:Δx _(j)=(X _(j) −X _(j,0))/X _(j,0)  (Equation 8)Experimental measurement error is expressed as the concentrationdifference, Y(t)=[C_(E)(t)−C_(D)(t)], where C_(E)(t) and C_(D)(t)represent the NO exhalation profiles predicted by the model and observedin hypothetical measurements, respectively. We assume that Y(t) is aGaussian white noise sequence with zero mean and variance, σ_(ED) ²,which results from random baseline fluctuations with a nominal standarddeviation of σ_(ED)=±1 ppb. The observed data is assumed to be sampledat 50 Hz, corresponding to a sampling time, T_(s)=0.02 s. Thus, for eachexhalation, m, we can define the discrete time difference,t−t_(0,m)−t_(I,m),=n T_(s) (n=0, 1, 2, . . . , N_(m)), where the totalnumber of sampled concentrations for tidal breath, m, is:N_(m)=t_(E,m)/T_(s). Hence, for each breath, we represent Y(t) as adiscrete sequence, Y(n), comprised of N_(m) independent and normallydistributed random variables. Thus, in general, a sequence of M breathsincludes L=Summation from m=1 to M of N_(m) independent and normallydistributed samples, each with variance σ_(ED) ², and zero mean.Sensitivity, an Uncertainty Analysis

We characterize the accuracy of a particular, flow-independent parameterestimate as the 90% confidence interval of parameter, X_(j), around itsfitted value, X_(j,0), with the other parameters, X_(i) (i≠j), fixed attheir fitted values, X_(i,0). Statistically, we define the 90%confidence interval, X_(j,u)≧X_(j)≧X_(j,L), as the range of variation inX_(j) around X_(j,0), over which there is 90% probability that X_(j)does not influence the error, Y(n). Thus, X_(j,u) and X_(j,L) are theupper and lower limits of X_(j) at 90% probability.

If a single parameter, X_(j), is varied around X_(j,0), with the otherparameters fixed at their fitted values, X_(i)=X_(i,0) (i≠j), thenC_(E)(n)=C_(E)(n, X_(j), X_(i,0)) is a function of the flow-independentparameters, which for the best unbiased estimate, X_(j)=X_(j,0), is:C_(E,0)(n)=C_(E)(n, X_(j,0), X_(i,0)). However, the sequence,Y₀(n)=[C_(E,0)(n, X_(j,0), X_(i,0))−C_(D)(n)], is a random variable,whereas Y(n)−Y₀(n)=[C_(E)(n, X_(j), X_(i,0))−C_(E,0)(n, X_(j,0),X_(i,0))] is a predetermined function of the model parameters and time,t=n Ts+t_(0,m)+t_(I,m). With these assumptions, we may estimateparameter confidence intervals, based on a simple t-test, for knownvariance, σ_(ED) ²:

$\begin{matrix}{{\sum\limits_{n = 1}^{L}\;\left\lbrack {{C_{E}\left( {n,{X_{i,}X_{i,0}}} \right)} - {C_{E,0}\left( {n,X_{j,0},X_{i,0}} \right)}} \right\rbrack^{2}} = {\sigma_{ED}^{2}{T\left( {L - P} \right)}^{2}}} & \text{(Equation 9)}\end{matrix}$where T(L−P) is the critical t-value at 90% confidence with L−P degreesof freedom, and where X_(j,0) is the hypothetical fitted value offlow-independent parameter and X_(j) is its value at a 90% confidencelimit (i.e., either X_(j,u) or X_(j,L)).

If C_(E)(n, X_(j), X_(i,0)) is a linear function of X_(j), forX_(j,u)≧X_(j)≧X_(j,L), then the relationship,ΔX_(j)=X_(j,u)−X_(j,0)=X_(j,0)−X_(j,L), holds and the 90% confidencelimit of X_(j) is expressed in terms of its fractional uncertainty,Δx_(j)=ΔX_(j)/X_(j,0). This is valid whenever C_(E)(n, X_(j), X_(i,0))is approximately linear in X_(j) around X_(j,0). In this case:Y(n)−Y₀(n)=S_(j)(n) ΔX_(j), where S_(j)(n) is the sensitivity ofC_(E)(n, X_(j), X_(i,0)) with respect to X_(j). We also define thenormalized or relative sensitivity, S^(r) _(j), and the semi-relativesensitivity, S^(sr) _(j), which represent the fractional and absolutechange of C_(E) per fractional change in X_(j), respectively. Thesethree quantities are related to each other as shown below:

$\begin{matrix}{{S_{j}(n)} = {{\frac{\partial{C_{E}\left( {n,X_{j},X_{i,0}} \right)}}{\partial X_{j}}\mspace{20mu}{where}\mspace{14mu} X_{j}} = {X_{j,0}\mspace{50mu} = {\frac{{C_{E}\left( {n,X_{j,0},X_{i,0}} \right)}{S_{j}^{r}(n)}}{X_{j,0}} = \frac{S_{j}^{\;{sr}}(n)}{X_{j,0}}}}}} & \text{(Equation 10)}\end{matrix}$Thus, if C_(E)(n, X_(j), X_(i,0)) is a linear function of X_(j),Equation 9 reduces to:

$\begin{matrix}{\left. {\left. {\Delta\; X_{j}{\overset{\_}{S}\left\lbrack {S_{j}(n)} \right\rbrack}^{2}} \right\}^{1/2} = {\Delta\; x_{j}{\sum\limits_{n = 1}^{L}\;\left\lbrack {S_{j}^{\;{sr}}(n)} \right\rbrack^{2}}}} \right\}^{1/2} = {{\pm \sigma_{ED}}{T\left( {L - P} \right)}}} & \text{(Equation 11)}\end{matrix}$

Since S^(sr) _(j)(t) is a function of time, we define the time-averaged,root-mean-squared semi-relative sensitivity,

${{\overset{\_}{S}}_{j}^{\;{{sr},{r\; m\; s}}}},$by averaging over each exhalation time interval, (t=t_(0,m)+t_(I,m) tot_(0,m)+t_(I,m)+t_(E,m)):

$\begin{matrix}{{{\overset{\_}{S}}_{j}^{\;{{sr},{r\; m\; s}}}}^{2} = {\left( \frac{1}{\left( {L - P} \right)} \right){\sum\limits_{n = 1}^{L}\;\left\lbrack {S_{j}^{\;{sr}}(n)} \right\rbrack^{2}}}} & \text{(Equation 12)}\end{matrix}$The quantity, (L−P), appears in Equation 12 as a correction for degreesof freedom. Thus, if | S _(j) ^(sr,rms)| is sufficiently large, ΔX_(j)is small and X_(j) can be accurately determined, or if | S _(j)^(sr,rms)| is small, ΔX_(j) is large. Thus, we express the 90%confidence intervals for individual, parameter hypotheses in terms oftheoretical fractional uncertainties by combining Equations 11 and 12 toobtain:

$\begin{matrix}{{\Delta\; X_{j}} = {{\pm \sigma_{E}}{{T\left( {L - P} \right)}/\left\{ {\left\lbrack {L - P} \right\rbrack^{1/2}{{\overset{\_}{S}}_{j}^{\;{{s\; r},{r\; m\; s}}}}} \right\}}}} & \text{(Equation 13)}\end{matrix}$Equation 13 is valid if C_(E)(n, X_(j), X_(i,0)) can be expressed as alinear function of X_(j) on the interval, X_(j,u)≧X_(j)≧X_(j,L). Ourresults suggest that this is true for X_(j)=D_(awNO), J′_(awNO) andC_(alv,ss). However, {circumflex over (D)}_(alvNO), V_(aw) and V_(ds)exhibit non-linear behavior and their confidence intervals are computedfrom Equation 9, for these flow-independent parameters. For the lattercase, unless otherwise indicated, Δx_(j)=ΔX_(j)/X_(j,0) is evaluated forΔX_(j) set equal to the maximum of X_(j,u)−X_(j,0) or X_(j,0)−X_(j,L).

The assumption of independent and identically distributed randomvariables is not valid if there is systematic deviation between theobserved data and predictions of the model. Nonetheless, the methodologypresented above yields preliminary estimates for parameteruncertainties, which can be used to design experimental protocols.

We computed the uncertainties of parameter estimates by using thet-statistic set forth above. As discussed above, theoretical exhalationprofiles are linear functions of J′_(awNO) and C_(alv,ss), and may beapproximated as a linear function of D_(awNO). Thus, we computed theirfractional uncertainties from Equations 12 and 13 above. However,D_(alvNO), V_(aw) and V_(ds) exhibit non-linear behavior; thus, wecomputed Δx_(j) from Equation 9 above for these parameters. We analyzedthe impact of the parameter value, the number of observed tidal breaths,and the breathing pattern on the confidence intervals.

Effect of Flow-independent Parameter Values

Flow-independent parameters demonstrate significant inter-subjectvariability, and thus the confidence interval for a given parameter mayvary. We studied the impact of the parameter value itself on theuncertainty by individually varying each parameter with the otherparameters fixed at their central values. Thus, we performed oursimulations with pre-selected central, lower and upper limit values forthe flow-independent parameters (see the table of FIG. 7). The set ofparameter values used for each simulation correspond to the hypotheticalbest, unbiased estimates, X_(j,0), determined experimentally.

Effect of the Number of Observed Tidal Breaths

A potential advantage of tidal breathing relative to the single breathmaneuver is the ability to easily observe multiple consecutive breathswhich may reduce the uncertainty in the estimated parameter value.Equations 9, 12 and 13 predict that Δx_(j) decreases as the total numberof samples, L, increases. Specifically, Δx_(j)→0, as L→∞. Thus,parameter uncertainties, which are exclusively the result of white noise(random errors), vanish for a large number of samples. Herein, we do notaccount for potential systematic errors (e. g., The variation of{circumflex over (D)}_(alvNO), resulting from its dependence uponalveolar volume, or ΔV_(alv), as discussed above). Unlike parameteruncertainties resulting from random errors, uncertainties resulting fromsystematic errors will not necessarily vanish for a large number ofsamples.

Effect of Breathing Pattern

Sustainable tidal breathing requires a minimum alveolar ventilationrate, {dot over (V)}_(alv), to supply oxygen to metabolizing tissue,which we specify as: {dot over(V)}_(alv)=[ΔV_(alv)−V_(aw)−V_(ds)]/(t_(I)+t_(E))=5,000 ml/min, whereΔV_(alv)=Q_(I)t_(I)=[V_(alv,I)−V_(alv,E)] is the tidal volume change(i.e., equivalent to the change in alveolar volume). For identicalbreaths (governed by Equations 3 to 6), the breathing pattern iscompletely characterized by the flow rate ratio,q=Q_(I)/Q_(E)=t_(E)/t_(I), and the breathing frequency, f_(B)=q/[(1+q)t_(E)]. Thus, f_(B) and q were varied at fixed {dot over (V)}_(alv) tospecify the breathing pattern. We specify central values q=2 andf_(B)=0.2 s⁻¹=12 min⁻¹, which correspond to: ΔV_(alv)=992 ml,Q_(E)=207.5 ml/s, Q_(I)=415 ml/s, t_(E)=3.33 s, and t_(I) =1.67 s. Byvarying q and f_(B) around their central values, we can identifybreathing patterns, which minimize the uncertainties of flow-independentparameter estimates.

As a basis for this analysis, we impose the upper limit: V_(alv,I)≦5,000ml, and specify the initial alveolar volume, V_(alv,E)=2,300 ml, whichcorresponds to a maximum tidal volume change ofΔV_(alv,max)=[V_(alv,I)−V_(alv,E)]_(max)=2,700 ml. For identical breathsthis specifies a lower limit for the breathing frequency, f_(B)≦{dotover (V)}_(alv)(ΔV_(alv,max)−V_(aw)−V_(ds)), which at the centralvalues, V_(aw)=200 ml and V_(ds)=75 ml, becomes: f_(B)≧(5,000ml/min)/(2,425 ml)≈2 breaths/min. ΔV_(alv) is constrained further byrequiring: τ_(Ia)+τ_(Ids)≦t_(I) and τ_(Ea)+τ_(Eds)≦t_(E).

Results

Effect of Flow-independent Parameter Values

We computed 90% confidence intervals for each flow-independent parameterafter one minute of tidal breathing by individually varying eachparameter, while fixing the other parameters at their central values(see the table of FIG. 7). We kept the control variables, q and f_(B),at 2 and 12 min⁻¹, respectively, in this analysis. These results areshown in FIGS. 8 and 9.

FIG. 8 depicts 90% confidence intervals, computed for J′_(awNO),D_(awNO), C_(alv,ss), {circumflex over (D)}_(alvNO), V_(aw) and V_(da),in terms of their fractional uncertainties, Δx_(j), after one minute oftidal breathing with q=2 and f_(B)=12 min⁻¹. These results suggest thatJ′_(awNO), C_(alv,ss), V_(aw) and V_(ds) are easily estimated, sincetheir theoretical uncertainties are less than 10% after one minute.However, D_(awNO) and {circumflex over (D)}_(alvNO) are more difficultto determine (their uncertainties exceed 30% after one minute).

Note that Δx_(j) decreases with increasing values of J′_(awNO),D_(awNO), C_(alv,ss), V_(aw) and V_(ds) (FIG. 8). However, for each ofthese parameters the absolute value Δx_(j)=X_(j,0)Δx_(j) is nearlyconstant (see the graph of FIG. 8). In contrast, FIG. 8 suggests thatthe uncertainty of {circumflex over (D)}_(alvNO), increases as itscentral value increases, with the confidence interval at its upper limit({circumflex over (D)}_(alvNO)=1.0 s⁻¹) exceeding the mean by more than500-fold. The uncertainties of J′_(awNO), D_(awNO), C_(alv,ss), V_(aw)and V_(ds) are all approximately symmetric about their central values.In contrast, {circumflex over (D)}_(alvNO), exhibits non-linearbehavior, as evidenced by its highly skewed distributions.

FIGS. 9A–9F illustrate the effect of variation in each flow-independentparameter upon the exhalation profile, C_(E)(t), for a single tidalbreath. Variation in J′_(awNO), C_(alv,ss), V_(aw) and V_(ds) result insignificantly different exhalation profiles (FIGS. 9A, 9C, 9E and 9F).As J′_(awNO) increases, C_(E)(t) increases in both Phases II and III(FIG. 9A), whereas the effect of increasing C_(alv,ss) is to increaseC_(E)(t) in Phase III alone (FIG. 9C). As shown in FIGS. 9F and 9E, theprimary effects of increasing V_(ds), and V_(aw) are to delay the onsetof Phase II and Phase III, respectively. Variation in D_(awNO) and{circumflex over (D)}_(alvNO) have relatively little impact uponC_(E)(t) (FIGS. 9B and 9D, respectively). We omitted J′_(awNO), V_(aw)and V_(ds) from subsequent analysis, and studied D_(awNO) and{circumflex over (D)}_(alvNO) in more detail. We also retainedC_(alv,ss) for comparison purposes.

Effect of the Number of Observed Tidal Breaths

FIG. 10 shows the dependence of Δx_(j) upon total monitoring time foreach flow-independent parameter. In this analysis, we fixed allflow-independent parameters at their central values, with the controlvariables, q and f_(B), fixed at 2 and 12 min⁻¹, respectively. FIG. 10demonstrates that parameter estimates can be improved by observing moretidal breaths. After one minute, the uncertainty of D_(awNO) exceeds50%, but it is reduced to approximately 17% after ten minutes.Similarly, the uncertainty of C_(alv,ss) falls from approximately 2%after one minute to below 0.7% and 0.5% after 10 and 20 minutes,respectively, and for {circumflex over (D)}_(alvNO) from approximately300% to 100% and 30%, respectively. This approach is readily implementedin practice, since tidal breathing is simple to perform. Each of thecurves shown in FIG. 10 is approximately linear on a logarithmic scale,with slope ˜−0.5. Thus, if Δx_(j) is known after one minute, its valueafter 10 minutes can be estimated by dividing by the square root of 10.

Effect of Breathing Pattern

FIGS. 11A, 11B and 11C show the dependence of Δx_(j) upon breathingpattern (characterized by varying the control variables, q and f_(B))for D_(awNO), C_(alv,ss) and {circumflex over (D)}_(alvNO),respectively. We computed these results for one minute of tidalbreathing with all flow-independent parameters fixed at their centralvalues. Low f_(B) and high q (relatively rapid inhalation followed byvery slow, sustained exhalation) favors the accurate determinationD_(awNO) and C_(alv,ss). For one minute of tidal breathing, theuncertainty of D_(awNO) is reduced below 25% for f_(B)<6 breaths/min.and q>4 (see FIG. 11A). After one minute, the uncertainty of C_(alv,ss)is below 5% for nearly all physiologically relevant breathing patterns(see FIG. 11B).

The dependence of Δx_(j) upon breathing pattern for {circumflex over(D)}_(alvNO) is considerably more complex (see FIG. 11C). For anintermediate range of flow rate ratios, dependent on f_(B) (e. g.,approximately 2<q<10, for f_(B)>4 breaths/min), the uncertainty of{circumflex over (D)}_(alvNO) is very large. In fact, a “separatix”defines the boundary for {circumflex over (D)}_(alvNO) at whichΔx_(j)→∞. Furthermore, for example, after one minute with f_(B)=12breaths/min, we may achieve Δx_(j)<100% for {circumflex over(D)}_(alvNO) within two regions: q>20 and 1>q>0.2. If we reduce f_(B) to6 breaths/min, we achieve Δx_(j)<100%, for q>40 and 0.5>q>0.2 (see FIG.11C). Thus, in general, {circumflex over (D)}_(alvNO) is the mostdifficult parameter to estimate.

Discussion

We have utilized the two-compartment model to simulate NO gas exchangein tidal breathing, and assessed the estimation of flow-independentparameters. Our analysis suggests that J′_(awNO), C_(alv,ss), V_(aw) andV_(ds−), are easily characterized, whereas D_(awNO) and {circumflex over(D)}_(alvNO) are more difficult to determine (see FIGS. 8, 9, and 10).This is a consequence of the relatively low sensitivity of C_(E)(t) toD_(awNO) and {circumflex over (D)}_(alvNO) as compared to itssensitivity to J′_(awNO), C_(alv,ss), V_(aw) and V_(ds), (see FIG. 9).

FIGS. 11A and 11B suggest that accurate determination of D_(awNO) and{circumflex over (D)}_(alvNO) are favored by relatively short durationinhalations (t_(I)=1 to 5 sec) followed by slow, long durationexhalations (t_(E)=6 to 12 sec). In addition, estimation of {circumflexover (D)}_(alvNO) is also favored at higher breathing frequency witheither slower inhalation or very rapid inhalation and very slowexhalation. Small values of ΔV_(alv) minimize the impact of systematicerrors on estimates of {circumflex over (D)}_(alvNO). Theoretically,breathing patterns, where f_(B)<4 breaths/min and q>10, are favorable toestimate both D_(awNO) and {circumflex over (D)}_(alvNO). However, thiscorresponds to t_(I)<1.4 sec, t_(E)>1 sec, and a high tidal volumechange (ΔV_(alv)˜1,500 ml), which may be difficult to achieve inpractice. Furthermore, such a high tidal volume change would also resultin a large systematic error (˜22%) for {circumflex over (D)}_(alvNO);which is not accounted for in this analysis. The remaining parameters,J′_(awNO), C_(alv,ss), V_(aw), and V_(ds), are readily estimated for anybreathing pattern.

For identical breaths, the exhalation profile for the two-compartmentmodel, C_(E)(t), is predicted by Equations 3 to 5. During Phase I(Equation 3), C_(E)(t)=O, and the dead space volume, V_(ds), constrainsthe duration of Phase I. However, Phase I does not provide anyinformation for estimation of the other flow-independent parameters.During Phase II (Equation 4), C_(E)(t) is independent of C_(alv,ss) and{circumflex over (D)}_(alvNO), but does provide information fordetermination of J′_(awNO) and D_(awNO). In addition, the duration ofPhase II is constrained by V_(aw). Thus, V_(ds) and V_(aw) aredetermined primarily by the relative lengths of the Phase I and Phase IItime intervals. During Phase III (Equation 5), C_(E)(t) depends upon allof the flow-independent parameters, and usually exhibits its maximumsensitivity to J′_(awNO), C_(alv,ss), D_(awNO) and {circumflex over(D)}_(alvNo) (see FIG. 9).

For single-breath maneuvers, such as pre-expiratory breathhold, Phase IIplays a major role in determining D_(awNO) and J′_(awNO). Accumulationof NO in the airway during breathhold leads to a marked peak of NO,which is observed in expired breath during Phase II. However, for tidalbreathing a single breath is completed within a much shorter timeinterval. Hence, much lower levels of NO accumulate in the airwaycompartment prior to exhalation, and, in most cases, an NO peak is notobserved during Phase II.

An element of air appearing in expired breath during Phase II, existedwithin the airway compartment, at some position V<V_(aw), at the startof exhalation. Thus, its residence time in the airway compartment is:(V_(aw)−V)/Q_(E)<τ_(Ea)=V_(aw)/Q_(E). However, an air element, whichappears in expired breath during Phase III, originated within thealveolar compartment, and its residence time in the airway compartmentis τ_(Ea). Thus, the residence time of expired air is longer in PhaseIII than in Phase II, which results in greater sensitivity of C_(E)(t)to the airway parameters, D_(awNO) and J′_(awNO), on the Phase III timeintervals. Therefore, for fixed monitoring time, optimal estimates forD_(awNO), J′_(awNO), C_(alv,ss) and {circumflex over (D)}_(alvNO), aredetermined by maximizing the Phase III exhalation interval for tidalbreathing (i.e., short duration inspiration with longer durationexpiration as described previously), since the sensitivity of C_(E)(t)to all of these parameters is maximum on this interval.

FIGS. 11A and 11B show that Δx_(j) decreases with increasingq=Q_(I)/Q_(E)=t_(E)/t_(I) and decreasing f_(B) for D_(awNO) andC_(alv,ss), which implies that Δx_(j) decreases as t_(E) and Q_(I)increase, and as Q_(E) decreases. Therefore, short duration inhalationat high flow followed by long duration inhalation at low flowfacilitates estimation of these parameters. Long exhalation times at lowflow allow higher NO concentrations to accumulate in the airway. Higherairway concentration increases the exhalation profile's sensitivity toD_(awNO). Thus, the uncertainty of D_(awNO) is minimized under theseconditions.

Short inhalation times at high flow transport less NO into the alveolarcompartment from the airway compartment, which leads to enhancedsensitivity of the Phase III exhalation profile to C_(alv,ss). Inaddition, greater ΔV_(alv) allows significant amounts of air from thealveolar compartment to reach the mouth, further increasing thesensitivity of the Phase III exhalation profile to C_(alv,ss). Atconstant alveolar ventilation rate (specified herein as {dot over(V)}_(alv)=5,000 ml/min), ΔV_(alv) is inversely proportional to f_(B).Therefore, lower breathing frequencies result in improved estimates forC_(alv,ss).

At very low breathing frequencies (f_(B)<5 breaths/min, see FIG. 11C),similar dynamics result in conditions favorable to estimates of{circumflex over (D)}_(alvNO). Long exhalation times at low flow buildup an appreciable amount of NO in the airway, which is transported intothe alveolar region on a subsequent (rapid) inhalation. Unfortunately,such low breathing frequencies are difficult to achieve in practice.However, at slightly higher breathing frequencies (5<f_(B)<8breaths/min), the uncertainty of {circumflex over (D)}_(alvNo) decreaseswith increasing f_(B), and is minimized in two distinct regions at veryhigh or low values of q. For example, with f_(B)=12 breaths/min, weachieve Δx_(j)<100%, for q>20 and 1>q>0.2 (see FIG. 11C). Under theseconditions {circumflex over (D)}_(alvNO) becomes very sensitive to theconcentration difference, (C_(alv,I)−C_(alv,ss)), and we can approximateΔx_(j) as inversely proportional to the absolute value,|C_(alv,I)−C_(alv,ss)|.

For q<1, the time duration of inhalation exceeds that for exhalation(t_(I)>t_(E)), and more of the NO accumulated in the airway istransported into the alveolar region during inhalation than is removedduring exhalation. Thus, C_(alv,I)>C_(alv,ss), which provides a gradientfor NO transport from airspace to tissue in the alveolar compartment,and increases the sensitivity of C_(E)(t) to {circumflex over(D)}_(alvNO). Thus, at any fixed breathing frequency, we can determine acritical flow rate ratio, where |C_(alv,I)−C_(alv,ss)| is maximized(e.g., q˜0.5 for f_(B)=12 breaths/min, see FIG. 11C). As q increases, asecond critical flow rate ratio is reached where C_(alv,I)=C_(alv,ss),and {circumflex over (D)}_(alvNO) cannot be determined (q˜15 forf_(B)=12 breaths/min, see FIG. 11C). As q increases further,C_(alv,I)<C_(alv,ss), and |C_(alv,I)−C_(alv,ss)| increases. Thus, lessof the NO accumulated in the airway is transported into the alveolarregion during inhalation than is removed during exhalation. This effectagain provides a gradient for NO transport from tissue to airspace inthe alveolar compartment, which increases the sensitivity of C_(E)(t) to{circumflex over (D)}_(alvNO). Therefore, improved estimates of{circumflex over (D)}_(alvNO) are favored when C_(alv)(t) becomes moredependent upon its history and significantly deviates from C_(alv,ss).Furthermore, these results suggest that estimates of {circumflex over(D)}_(alvNO) should be strongly dependent upon C_(alv,ss).

We have not addressed the potential impact of systematic errors uponparameter estimates from experimental data, such as the dependence of{circumflex over (D)}_(alvNO) upon V_(alv). Large tidal volume changesmay adversely affect estimation of {circumflex over (D)}_(alvNO), sinceincreasing ΔV_(alv)/V_(alv) results in greater variation of {circumflexover (D)}_(alvNO) over the time course of exhalation. Additionalsystematic errors may be introduced by the finite response time of theanalytical monitoring system. Time lags, resulting from such limitationsas finite instrument response time, transit times in instrumentplumbing, etc., are negligible for single breath maneuvers. However,time lags are more important for tidal breathing, due to the shortertime duration of each breath. Imprecise modeling of time lags may resultin miss-alignment of experimental concentration and flow rate profiles,thereby causing incorrect placement of the Phase I, II and III timewindows. Thus, precise characterization of system time lags is necessaryto facilitate accurate parameter estimates from tidal breathing data.These time lags are dependent upon V_(aw) and V_(ds), which have notbeen estimated from experimental data in previous efforts. Fortunately,the results presented herein suggest that V_(ds) and V_(aw) are readilydetermined from experimental data, since these two parameters dependupon the time durations of the Phase I and Phase II intervals,respectively.

CONCLUSIONS

Our results are based on the central values of the flow-independentparameters, shown in the table of FIG. 7. If the flow-independentparameters deviate significantly from their central values, appropriatemodifications to this protocol are readily determined using themethodology presented. Our analysis suggests that J′_(awNO), C_(alv,ss),V_(ds), and V_(aw) should be readily determined from tidal breathing;however, D_(awNO) and D_(alvNO) are more difficult to estimate, and mayrequire multiple tidal breathing patterns, or a relatively large numberof tidal breaths. In addition, a short inspiration time relative toexpiration reduces the uncertainty for all of the flow-independentparameters. We conclude that a tidal breathing pattern has the potentialto characterize flow-independent NO exchange parameters.

NOMENCLATURE AND ABBREVIATIONS

-   C_(air)=C_(air)(t,V)=NO concentration in the airway gas space (ppb).-   C_(air,end) C_(air)(t,V=O) for inhalation, and C_(air,end)=C_(alv)    for exhalation (ppb).-   C_(alv)(t)=gas phase NO concentration in alveolar compartment (ppb).-   C_(alv,0)=C_(alv,E)=C_(alv)(t=0)=final (exhalation) alveolar    concentration at t=0 (ppb).-   C_(alv,I)=C_(alv)(t=t_(I))=final (inhalation) alveolar concentration    at t=t_(I) (ppb).-   C_(alv,ss)=steady state alveolar concentration (ppb).-   C_(awNO)=J′_(awNO)/D_(awNO)=equivalent gas phase, airway tissue    concentration (ppb).-   C_(D)(t) C_(D)(nT_(s))=observed (experimental) NO exhalation profile    (signal) (ppb).-   C_(E)(t) NO concentration at mouth (exhalation profile) (ppb).-   D_(alvNO)=diffusing capacity of NO in the alveolar region (ml/s).-   {circumflex over (D)}_(alvNO)=D_(alvNO)/V_(alv)=alveolar diffusing    capacity per unit alveolar volume (s⁻¹).-   D_(awNO)=airway diffusing capacity (ml−s⁻¹-ppb⁻¹ or pl-s⁻¹-ppb⁻¹).-   {circumflex over (D)}_(awNO)=D_(awNO)/V_(aw).-   f_(a)=f_(a1)+f_(a2).-   f_(a1)=[1−e^(−{circumflex over (D)}) ^(awNO) ^(t) ^(I) ^((τ) ^(Ea)    ^(+τ) ^(Ia)) ^(−{circumflex over (D)}) ^(alvNO() ^(t) ^(I) ^(−τ)    ^(Ia) ^(−τ) ^(Ids)) ]/{circumflex over (D)}_(alvNO).-   f_(a2)={circumflex over (D)}_(awNO)(1+q)[e^(−{circumflex over (D)})    ^(alvNO) ^(t) ^(I) −e^(−{circumflex over (D)}) ^(awNO) ^(t) ^(I)    ^((τ) ^(Ea) ^(+τ) ^(Ia)) ^(−{circumflex over (D)}) ^(alvNO) ^((t)    ^(I−) ^(τ) ^(Ia)) ]/[{circumflex over (D)}_(alvNO)({circumflex over    (D)}alvNO−{circumflex over (D)}_(awNO))(1+q)].-   f_(B)q/[(I+qt_(E))]=breathing frequency.-   f_(t)=f_(a1)+f_(a2)+[e^(−{circumflex over (D)}) ^(awNO) ^(τ) ^(Ia)    ^(−{circumflex over (D)}) ^(alvNO) ^((t) ^(I) ^(−τ) ^(Ids) ^(−τ)    ^(Ia)) −e^(−{circumflex over (D)}) ^(awNO) ^(τ) ^(Ia) ]/{circumflex    over (D)}_(alvNO).-   f_(QI)=f_(QI1)+f_(QI2)−f_(QI3).-   f_(QI1)=e^(−{circumflex over (D)}) ^(awNO) ^((τ) ^(Ea) ^(+τ) ^(Ia))    ^(−{circumflex over (D)}) ^(alvNO) ^((t) ^(E) ^(+t) ^(I) ^(−τ)    ^(Eds) ^(−τ) ^(Ea) ^(−τ) ⁾ /[(1+q){circumflex over (D)}_(alv)].-   f_(QI2)=e^(−{circumflex over (D)}) ^(awNO) ^((τ) ^(Ea) ^(+τ) ^(Ia))    ^(−{circumflex over (D)}) ^(alvNO) ^((t) ^(E) ^(+t) ^(I) ^(−τ) ^(Ea)    ^(−τ) ^(Ia)) /[(1+q) ({circumflex over (D)}_(alvNO)({circumflex over    (D)}_(alvNO)−{circumflex over (D)}_(awNO))].-   f_(QI3)=e^(−{circumflex over (D)}) ^(alvNO) ^((t) ^(I) ^(+t) ^(E))    /[(1+q) ({circumflex over (D)}_(alvNO)−{circumflex over    (D)}_(awNO))].-   FRC, functional reserve capacity.-   j=index for flow-independent parameters (j=1, . . . , P=6).-   J′_(alvNO)=global maximum flux of NO in alveolar compartment    (ml−ppb/s).-   (J′_(alvNO) is defined as the flux of NO into the alveolar    compartment, if C_(alv)(t)=0).-   J_(awNO)=net flux of NO into the air space of the airway compartment    (pl/s or ml/s).-   J_(awNO)=J′_(awNO)−D_(awNO)C=D_(awNO)[C_(awNO)−C].-   J′_(awNO)=maximum volumetric airway flux (pl/s or ml/s).-   J_(alvNO)=net flux of NO into the air space of the alveolar    compartment (ml−ppb/s).-   J_(alvNO)=J′_(alvNO)−D_(alvNO)C_(alv)(t)=D_(alvNO)[C_(alv,ss)−C_(alv)(t)].-   L=Summation from m=1 to M of N_(m)=total number of exhalation    profile measurements in M breaths.-   m=index for tidal breaths, m=1, 2, . . . , M.-   M=total number of tidal breaths observed in an exhalation profile    sequence.-   n=index for data samples=(t−t_(I))I T_(s), n=0, 1, 2, . . . N_(m);    in tidal breath, m.-   N_(m)=number of sampled concentrations in tidal breath, m.-   pl, picoliter.-   P_(m)=Q_(E,m)/Q_(E,m−1)-   ppb, parts per billion.-   P=Number of fitted flow-independent parameters (P=6).-   q=Q_(I)/Q_(E)=t_(E)/t_(I)=flow rate ratio.-   q₁=(Q_(I)/Q_(E))₁=(t_(E)/t_(I))₁, flow rate ratio for the first    observed breath (m=1).-   q_(m)=(Q_(I)/Q_(E))_(m)=(t_(E)/t_(I))_(m), flow rate ratio for    breath, m.-   Q=volumetric, air flow rate=−Q_(I)(inhalation), Q_(E)(exhalation)    (ml/s).-   Q_(E)=air flow rate averaged over exhalation time interval (ml/s).-   Q_(E,m)=average exhalation air flow rate for breath, m (ml/s).-   Q_(I)=air flow rate averaged over inhalation time interval (ml/s).-   Q_(I,m)=average inhalation air flow rate for breath, m (ml/s).-   rms, root-mean-squared.-   S_(i,j)=sensitivity of an output, i, to an input, j.-   S^(r) _(i,j)=normalized or relative sensitivity of an output, i, to    an input, j.-   S^(sr) _(i,j)=semi-relative sensitivity of an output, i, to an    input, j.-   | S ^(sr,rm){circumflex over (D)}_(alvNO) ^(s)c,j|=time-averaged    (rms) semi-relative sensitivity of Y_(i) to X_(j).-   t=time (s).-   t_(E) exhalation time interval (s).-   t_(I) inhalation time interval (s).-   Ts=sampling time of observed concentration data=0.02 s (50 Hz).-   V=axial position in units of cumulative volume (ml).-   V_(alv)(t)=alveolar compartment volume (ml).-   V_(alv,E)=V_(alv)(t=0)={V_(alv)(t=t_(I)+t_(E))= . . . for identical    breaths} (ml).-   V_(alv,I)=V_(alv)(t=t_(I))={V_(alv)(t=2t_(I)+t_(E))= . . . for    identical breaths} (ml).-   V_(aw)=airway compartment volume (ml).-   V_(ds)=dead space compartment volume (ml).-   X_(j)=variation in flow-independent parameter, j, around its fitted    value, X_(j,0).-   X_(j,0)=best unbiased estimate fitted value of flow-independent    parameter, j.-   Y(n)=[C_(E)(n)−C_(D)(n)]=discrete time experimental measurement    error (ppb).-   Y(t)=[C_(E)(t)−C_(D)(t)]=experimental measurement error (ppb).-   ΔV_(alv,max)=[V_(alv,I)−V_(alv,0)]_(max)=maximum tidal volume change    (ml).-   ΔV_(alv)=tidal volume change (ml).-   Δx_(j)=fractional uncertainty of flow-independent parameter, j.-   ΔX_(j)=uncertainty of flow-independent parameter, j.-   τ_(E)=V/Q_(E)=exhalation residence time (s).-   τ_(Ea)=V_(aw)/Q_(E)=exhalation residence time for the airway    compartment (s).-   τ_(Eds)=V_(ds)/Q_(E)=exhalation residence time for the dead space    compartment (s).-   τ_(I)=V/Q_(I)=inhalation residence time (s).-   τ_(Ia)=V_(aw)/Q_(I)=inhalation residence time for the airway    compartment (s).-   τ_(Ids)=Vd_(ds)/Q_(I)=inhalation residence time for the dead space    compartment (s).

Many alterations and modifications may be made by those having ordinaryskill in the art without departing from the spirit and scope of theinvention. Therefore, it must be understood that the illustratedembodiment has been set forth only for the purposes of example and thatit should not be taken as limiting the invention as defined by thefollowing claims. For example, notwithstanding the fact that theelements of a claim are set forth below in a certain combination, itmust be expressly understood that the invention includes othercombinations of fewer, more or different elements, which are disclosedin above even when not initially claimed in such combinations.

The words used in this specification to describe the invention and itsvarious embodiments are to be understood not only in the sense of theircommonly defined meanings, but to include by special definition in thisspecification structure, material or acts beyond the scope of thecommonly defined meanings. Thus if an element can be understood in thecontext of this specification as including more than one meaning, thenits use in a claim must be understood as being generic to all possiblemeanings supported by the specification and by the word itself.

The definitions of the words or elements of the following claims are,therefore, defined in this specification to include not only thecombination of elements which are literally set forth, but allequivalent structure, material or acts for performing substantially thesame function in substantially the same way to obtain substantially thesame result. In this sense it is therefore contemplated that anequivalent substitution of two or more elements may be made for any oneof the elements in the claims below or that a single element may besubstituted for two or more elements in a claim. Although elements maybe described above as acting in certain combinations and even initiallyclaimed as such, it is to be expressly understood that one or moreelements from a claimed combination can in some cases be excised fromthe combination and that the claimed combination may be directed to asubcombination or variation of a subcombination.

Insubstantial changes from the claimed subject matter as viewed by aperson with ordinary skill in the art, now known or later devised, areexpressly contemplated as being equivalently within the scope of theclaims. Therefore, obvious substitutions now or later known to one withordinary skill in the art are defined to be within the scope of thedefined elements.

The claims are thus to be understood to include what is specificallyillustrated and described above, what is conceptionally equivalent, whatcan be obviously substituted and also what essentially incorporates theessential idea of the invention.

1. A method of characterizing nitric oxide exchange in lungs duringtidal breathing, comprising: utilizing a two compartment model, whereina first compartment represents an alveolar region of the lungs and asecond compartment represents an airway to the lungs; selecting severalflow-independent parameters; and estimating the values of the parametersby utilizing the two compartment model fitted to previously obtainedexperimental data gathered during tidal breathing.
 2. The method ofclaim 1, wherein the flow independent parameters include one or more ofJ′_(awNO), D_(awNO), C_(alv,ss), {circumflex over (D)}_(alvNO), V_(aw)and V_(ds).
 3. The method of claim 1, wherein the step of estimatingfurther comprises characterizing at least V_(aw) and V_(ds) of saidseveral flow-independent parameters based on the two-compartment model.4. The method of claim 1, wherein the step of estimating furthercomprises minimizing a difference between estimated values andexperimentally obtained values.
 5. The method of characterizing of claim1, further comprising: representing estimated values by a data curve oran equation defining the curve; wherein said step of estimatingcomprises determining a first confidence interval after a firstmonitoring time; and wherein said step of estimating further comprisesprojecting at least one of said values having improved accuracy and animproved second confidence interval after a second predeterminedmonitoring time, wherein the projection is based on the data curve orthe equation representing the curve.
 6. The method of characterizing ofclaim 1, wherein the step of estimating further comprises improving theaccuracy of the parameter values by reducing the frequency of tidalbreathing during monitoring from an initial value of the frequency oftidal breathing.
 7. The method of characterizing of claim 6, wherein thefrequency of breathing is reduced by increasing the time for each tidalbreath to in the range from 7 to 17 seconds.
 8. The method ofcharacterizing of claim 1, wherein the step of estimating furthercomprises improving the accuracy of the parameter values by increasingthe flow rate ratio q of the inhalation flow rate Q_(I), relative to theexhalation flow rate Q_(E) as compared to an initial magnitude of theratio q.
 9. The method of characterizing of claim 8, further comprisingincreasing the flow rate ratio q into the range from 6/5 to
 12. 10. Themethod of characterizing of claim 1, further comprising the preliminarysteps of: having a subject perform a tidal breathing maneuver; measuringand recording NO concentration and flow rate simultaneously during themaneuver; wherein the step of utilizing further comprises simulating thetidal breathing maneuver by the two-compartment model; the step ofestimating further comprises the preliminary step of fitting recordeddata to simulated data from the two-compartment model; wherein themethod further comprises the additional preliminary steps of: comparingthe recorded data and the simulated data; and selecting flow-independentparameters that by comparison to the measured and recorded data fallwithin a predetermined confidence interval.